Another important factor is the thickness of the wire. But, to properly account for this, you typically need brute force computing techiques--formulas become to hard to come by, even for highly skilled mathematicians and PHD level radio engineers who write those thick textbooks.

In the process of putting up a ~31' vertical over 36 radials, and in the process of understanding the remote tuner situation, I think I would like to be able to know what the matching impedances the tuner is going to face on different bands.

I have the AH-4 auto tuner which reputedly has a range of ~4 to ~5000 ohms. Is this enough? Icom says to avoid half wave wires, but DX Engineering sells a magic box that likely is a transformer to reduce the impedance to within an acceptable range.

If you have this ~31' wire, it seems to me it has one impedance at 7.1 Mhz, another at 14.2 or so, yet another at 21.3 etc. Is there a formula or table to ascertain what it is likely to be?

But this, my friend, is a piece of cake for EZNEC! I believe the free version would work just fine for your simple antenna. I would download it and play with it. http://www.eznec.com/demoinfo.htm

You can use a sample model of a vertical that comes with the program, change the wire length to 31', and it will create a table of the numbers that you need very, very easily across an entire band. If I were not so busy here, I would do it for you for 40, 20, and 15 meters, every 100 kHz on each band. Maybe someone else has more time, if you don't want to install EZNEC.

While you can use a 4:1 balun to reduce the impedance of a half wave wire, this will also reduce the impedance of the same wire used as a quarter wavelength. For example, that 31 ft wire may have an impedance around 40 ohms over that superb ground plane on 40 meters. A 4:1 balun would reduce that impedance to just 10 ohms--that sort of impedance may be too low for your tuner to operate as efficiently as you would like.

You could swap out the 4:1 balun to a 1:1 balun to operate the low bands more efficiently. Alternately, top loading can be used to raise the impedance, but this is normally too much of a hassle for multiband antennas, though I have found it works superbly for single band antennas.

Another possible option is to use multiple radiating elements with multiple tuners, but this may have too much of a visual impact for your situation. Luckily, my wife and neighbors don't seem to mind all the wires strung up over our flower beds--we have 200 big healthy rose bushes!

This is what creates the uncertainty. It would be worthwhile to eliminate the forbidden lengths, but the side effects might create worse problems than are solved. I hoped there was a formula, graph or table that might give some insight.

Unfortunately, although I have EZNEC by virtue of having the 22d Edition of Antenna Book, I do not have enough familiarity to use it..... one of the projects on my list once I get this antenna up and on the air.